173 research outputs found
On the magnitude of spheres, surfaces and other homogeneous spaces
In this paper we define the magnitude of metric spaces using measures rather
than finite subsets as had been done previously and show that this agrees with
earlier work with Leinster in arXiv:0908.1582. An explicit formula for the
magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary
homogeneous Riemannian manifold the leading terms of the asymptotic expansion
of the magnitude are calculated and expressed in terms of the volume and total
scalar curvature of the manifold. In the particular case of a homogeneous
surface the form of the asymptotics can be given exactly up to vanishing terms
and this involves just the area and Euler characteristic in the way conjectured
for subsets of Euclidean space in previous work.Comment: 21 pages. Main change from v1: details added to proof of Theorem
Euler-Bessel and Euler-Fourier Transforms
We consider a topological integral transform of Bessel (concentric
isospectral sets) type and Fourier (hyperplane isospectral sets) type, using
the Euler characteristic as a measure. These transforms convert constructible
\zed-valued functions to continuous -valued functions over a vector
space. Core contributions include: the definition of the topological Bessel
transform; a relationship in terms of the logarithmic blowup of the topological
Fourier transform; and a novel Morse index formula for the transforms. We then
apply the theory to problems of target reconstruction from enumerative sensor
data, including localization and shape discrimination. This last application
utilizes an extension of spatially variant apodization (SVA) to mitigate
sidelobe phenomena
The equation of state for two-dimensional hard-sphere gases: Hard-sphere gases as ideal gases with multi-core boundaries
The equation of state for a two-dimensional hard-sphere gas is difficult to
calculate by usual methods. In this paper we develop an approach for
calculating the equation of state of hard-sphere gases, both for two- and
three-dimensional cases. By regarding a hard-sphere gas as an ideal gas
confined in a container with a multi-core (excluded sphere) boundary, we treat
the hard-sphere interaction in an interacting gas as the boundary effect on an
ideal quantum gas; this enables us to treat an interacting gas as an ideal one.
We calculate the equation of state for a three-dimensional hard-sphere gas with
spin , and compare it with the results obtained by other methods. By this
approach the equation of state for a two-dimensional hard-sphere gas can be
calculated directly.Comment: 9 pages, 1 figur
О собственном времени очага сильного землетрясения
The physics of earthquakes was contriubuted to by the concept of proper time of the source of a strong earthquake, which is different from universal (calendar) time. The earlier idea of proper time was implicit and has been considered only in relation to the physics of aftershocks. The present paper extends the applicability of the concept of proper time, proposes a possible way of its measuring, and provides an example to illustrate the procedure for sequential ordering of earthquakes by proper time. The object of this study is a global activity of strong (M≥7) earthquakes. We consider the sequence of earthquakes as a Poisson-type random process. Comparatively weak earthquakes are used as the "underground clock", the tick of which marks the proper time. The Poisson distribution is compared with the distributions for two sequences of strong earthquakes. One of the sequences is ordered by universal time, and another - by proper time. The studies indicate the distribution of events ordered by proper time is closer to the Poisson distribution than that of events ordered by universal time. We attribute this to the non-stationarity of the geological medium, which is an immanent property of the Earth's lithosphere.В физику землетрясений введено понятие о собственном времени очага сильного землетрясения, отличном от универсального (календарного) времени. Ранее использовалась идея о собственном времени, но неявно и только лишь в узкой области, относящейся к физике афтершоков. В данной работе расширена область применимости представления о собственном времени, указан возможный способ его измерения и приведен пример, иллюстрирующий процедуру упорядочивания последовательности землетрясений в собственном времени. В качестве объекта исследования выбрана глобальная активность сильных землетрясений (М≥7). Последовательность землетрясений мы рассматриваем как случайный процесс пуассоновского типа. В качестве «подземных часов», тиканье которых отмечает ход собственного времени, использованы сравнительно слабые землетрясения. Распределение Пуассона сопоставлено с распределениями для двух последовательностей сильных землетрясений, одна из которых упорядочена по универсальному времени, а другая - по собственному. Результат испытания показал, что распределение событий, упорядоченных по собственному времени, ближе к распределению Пуассона, чем распределение событий, упорядоченных по универсальному времени. Авторы объясняют это нестационарностью геологической среды, которая является имманентным свойством литосферы Земли
Beyond genus statistics: a unifying approach to the morphology of cosmic structure
The genus statistics of isodensity contours has become a well-established
tool in cosmology. In this Letter we place the genus in the wider framework of
a complete family of morphological descriptors. These are known as the
Minkowski functionals, and we here apply them for the first time to isodensity
contours of a continuous random field. By taking two equivalent approaches, one
through differential geometry, the other through integral geometry, we derive
two complementary formulae suitable for numerically calculating the Minkowski
functionals. As an example we apply them to simulated Gaussian random fields
and compare the outcome to the analytically known results, demonstrating that
both are indeed well suited for numerical evaluation. The code used for
calculating all Minkowski functionals is available from the authors.Comment: 8 pages plus 1 figure; uses aaspp4.sty and flushrt.sty. Matches
version accepted for publication in Ap. J. Let
Emergence of Secondary Motifs in Tube-Like Polymers in a Solvent
We study the effects of two kinds of interactions in tube-like polymers and
demonstrate that they result in the formation of secondary motifs. The first
has an entropic origin and is a measure of the effective space available to the
solvent. The second arises from solvophobic interactions of the solvent with
the polymers and leads to an energy proportional to the contact surface between
the tube and solvent particles. The solvent molecules are modeled as hard
spheres and the two interactions are considered separately with the solvent
density affecting their relative strength. In addition to analytical
calculations, we present the results of numerical simulations in order to
understand the role played by the finite length of short polymers and the
discrete versus continuum descriptions of the system in determining the
preferred conformation.Comment: 5 pages, 2 figures, 1 table. Accepted by Phys. Rev.
Integral geometry of complex space forms
We show how Alesker's theory of valuations on manifolds gives rise to an
algebraic picture of the integral geometry of any Riemannian isotropic space.
We then apply this method to give a thorough account of the integral geometry
of the complex space forms, i.e. complex projective space, complex hyperbolic
space and complex euclidean space. In particular, we compute the family of
kinematic formulas for invariant valuations and invariant curvature measures in
these spaces. In addition to new and more efficient framings of the tube
formulas of Gray and the kinematic formulas of Shifrin, this approach yields a
new formula expressing the volumes of the tubes about a totally real
submanifold in terms of its intrinsic Riemannian structure. We also show by
direct calculation that the Lipschitz-Killing valuations stabilize the subspace
of invariant angular curvature measures, suggesting the possibility that a
similar phenomenon holds for all Riemannian manifolds. We conclude with a
number of open questions and conjectures.Comment: 68 pages; minor change
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
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